Classical and Spin Foam Quantum Gravity for Complex, Real Orthogonal
Groups and the Square
of Area Reality Constraint

Abstract:

Complex general relativity is the analytical continuation of the
SO(4,C) general
relativity on a real 4D manifold to a complex 4D manifold. By modifying the
Plebanski theory of
the SO(4,C)general relativity by adding a Lagrange multiplier to impose the
area metric reality
condition, the classical real general relativity theories for all signatures
can be derived. Spin
foams are path integral quantizations of discrete general relativity on
simplicial manifolds.
Various spin foam models for general relativity are available, the most
popular being the
Barrett-Crane models. The discrete form of the area metric on a simplicial
manifold is the square
of areas of the 2-simplices. In spin foams the Casimir values of the
representations that are used
to label the 2-simplices are defined as the square of the areas. The
discretized SO(4,C) general
relativity cannot be analytically continued to complex general relativity
but it can be spin foam
quantized. First I discuss the derivation of the Barrett-Crane model for
SO(4,C) general
relativity using a new direct procedure. Then by imposing the square of area
reality condition the
spin foams of real general relativity for all 4D signatures can be formally
deduced in an
intuitive way. By this procedure the Barrett-Crane models for all 4D
signatures and the SO(4,C)
general relativity are unified under a general picture. If time is available
I will discuss the
following ideas briefly: Relating spin foams to the spin network functionals
of canonical quantum
general relativity, the asymptotic limit of spin foams and the SO(4,C) Regge
Calculus as a general
form of Regge Calculus in 4D.